A polytope is fissary if it contains compound figures, such as vertex figures, edge figures, etc. A polyhedron is fissary when it has compound vertex figures. A polychoron is fissary when it has compound vertex or edge figures. A n-polytope is fissary if it has any compound m-dimensional figures where m ranges from 0 to n-3.

The only fissary uniform polyhedra are compounds. The first dimension to feature fissary uniform polytopes that are not compounds is 4, such as Sitphi (vertex fissary) and Dupti (edge fissary).

Fissary uniform polytopes may or may not be excluded from the total list of uniform polytopes in a given dimension, unlike exotic ones (having compound ridges or n-2 figures). However, fissaries dominate the uniform polytopes in high dimensions if allowed.